Civil Engineering Reference
In-Depth Information
and the stiffness matrix, is
k
(
e
)
=
[
B
]
T
[
D
][
B
]d
V
(
e
)
(9.98)
V
(
e
)
While Equation 9.98 is becoming rather familiar, a word or two of caution is
appropriate. Recall in particular that, although the interpolation functions used
here are two dimensional, the axisymmetric element is truly three dimensional
(toroidal). Second, the element is not a constant strain element, owing to the
inverse variation of
ε
with radial position, so the integrand in Equation 9.98 is
not constant. Finally, note that [
D
] is significantly different in comparison to the
counterpart material property matrices for plane stress and plane strain. Taking
the first observation into account and recalling Equation 6.93, the stiffness
matrix is defined by
k
(
e
)
=
[
B
]
T
[
D
][
B
]
r
d
r
d
z
2
(9.99)
A
(
e
)
and is a 6
×
6 symmetric matrix requiring, in theory, evaluation of 21 integrals.
Explicit term-by-term integration is not recommended, owing to the algebraic
complexity. When high accuracy is required, Gauss-type numerical integration
using integration points specifically determined for triangular regions [3] is used.
Another approach is to evaluate matrix [
B
] at the centroid of the element in an
rz
plane. In this case, the matrices in the integrand become constant and the stiff-
ness matrix is approximated by
k
(
e
)
≈
rA
[
B
]
T
[
D
][
B
]
2
¯
(9.100)
Of course, the accuracy of the approximation improves as element size is
decreased.
Referring to a previous observation, formulation of the [
B
] matrix is trouble-
some if
r
=
0 is included in the domain. In this occurrence, three terms of Equa-
tion 9.97 “blow up,” owing to division by zero. If the stiffness matrix is evaluated
using the centroidal approximation of Equation 9.100, the problem is avoided,
since the radial coordinate of the centroid of any element cannot be zero in an
axisymmetric finite element model. Nevertheless, radial and tangential strain and
stress components cannot be evaluated at nodes for which
r
=
0
.
Physically, we
know that the radial and tangential displacements at
r
=
0 in an axisymmetric
problem must be zero. Mathematically, the observation is not accounted for in
the general finite element formulation, which is for an arbitrary domain. One
technique for avoiding the problem is to include a hole, coinciding with the
z
axis
and having a small, but finite radius [4].
9.5.2 Element Loads
Axisymmetric problems often involve surface forces in the form of internal or
external pressure and body forces arising from rotation of the body (centrifugal
